Pre-requisites:

CHE311, ESO204, ME231, or equivalent (recommended)

Course Content:

This course aims to introduce and study the different instabilities observed in the field of fluid mechanics. The tools used for analysing the stability of different fluid flows shall be covered along with pertinent research topics. Numerical methods relevant in the field of stability of fluid flows are discussed wherever it may be necessary.

Lecturewise Breakup (based on 50min per lecture)

I. Phenomenology: (2 Lectures)

• A brief overview of different instabilities observed in nature, historical perspective.

II. The concept of stability: (3 Lectures)

• Stability in the sense of Lyapunov and other approaches, normal modes, stability-instability of steady states using ODEs.

III. The rotating pendulum problem: (4 Lectures)

• Base-states, linearised system of equations, stability analysis, preliminary investigation using numerical methods.

IV. Kelvin-Helmholtz instability: (5 Lectures)

• Equations of motion, boundary conditions at the interface, linearised problem, effect of concentrated shear, influence of surface tension on the shear instability, Rayleigh-Taylor instability.

V. Instability of Taylor-Couette flow: (4 Lectures)

• Inviscid analysis, Rayleigh criterion, basic flow and stability analysis, viscous flow, introduction to primary and secondary instabilities.

VI. Rayleigh-Bénard instability: (5 Lectures)

• Oberbeck-Boussinesq equations, pure conduction solution, linearised system and normal mode expansion, effect of boundary conditions (isothermal, rigid surface, free surface), principle of exchange of stabilities.

VII. Capillary instability of a jet: (3 Lectures)

• Shape of a falling jet, Plateau-Rayleigh instability.

VIII. Introduction to weakly nonlinear analysis: (5 Lectures)

• Development of asymptotic expansions, introduction to multiple-time scales method, weakly damped oscillator, amplitude equation and Fredholm alternative.

IX. Introduction to bifurcation theory: (4 Lectures)

• Overview of different bifurcation scenarios, dynamics of strongly confined systems, Stuart-Landau-Watson equation, equilibrium solutions.

X. Introduction to instabilities in shear flows and non-modal stability analysis: (5 Lectures)

• Stability of parallel flows, Orr-Sommerfeld equation, Rayleigh's stability problem, absolute/convective instabilities, normal and non-normal systems, transient energy growth.

References:

1. Introduction to hydrodynamic stability, Philip G Drazin, Cambridge University Press, 2002
2. Hydrodynamic stability, Philip G Drazin, William H Reid, Cambridge University Press, 2004
3. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Steven H Strogatz, CRC Press, 2018
4. Open shear flow instabilities, Perspectives in Fluid Dynamics, Patrick Huerre, Editors: GK Batchelor, HK Moffatt, MG Worster, Cambridge University Press, 2000
5. Stability and transition in shear flows, P. J. Schmid, and D. Henningson, Springer Verlag, 2000